3 Climate dynamics

3.1: Matthews’ Approximation

For this, we follow the simplified climate dynamics used in Barnett et al. [4]. Their approach is based on an approximation from the geoscience literature used to support model comparisons. Specifically, Matthews Matthews et al. [5] (2009) and others have purposefully constructed an approximation for climate models outputs:

\[\text{ temperature anomaly } (Y_t) \approx \textrm{ TCRE} (\theta) \times \textrm{ cumulative emissions }\]

where TCRE is an acronym for the Transient Climate Response to cumulative Emissions. This simplified formulation abstracts from transitory ``weather’’ fluctuations in temperature. Instead emissions today have a long-lasting impact on temperature in the future where TCRE is a measure of climate sensitivity.

Our specific form:

\[\begin{equation} d Y_t = \mathcal{E}_t[\theta(m)d t+\varsigma d W_t^y] \end{equation}\]

for \(\theta(m) \in \Theta\) where \(\theta(m)\) is a specific TCRE obtained from the set of TCRE models \(\Theta\). Here we constructed this histogram. For baseline probabilities, we presume that each model has subjective probability \({\frac 1 M}\) and \(M\) is the number of climate models we use as inputs. In other words, we treat each such model as having the same subjective probability. We abstract from parameter learning since learning about such parameters has been very slow.

3.2 Approximation

More specifically, we use exponentially weighted average of each of response functions as coefficients \(\theta(m) \in \Theta\) in our computations, where the discount rate is set as \(\delta=0.01\) and the number of climate models is \(M = \# \Theta = 144\).

The histogram of these coefficients are represented below:

from src.plot import plot_hist
plot_hist("Figure 1: Histograms for climate sensitivity parameters")

3.3 Intensity function

The intensity function, \({\mathcal J}\), determines the possibility of a jump over the next small increment in time. For \(Y_t = y\), \(\epsilon \mathcal J (y)\) is the approximate jump probability over small time increment \(\epsilon\). Equivalently, \({\mathcal J}\) is a local measure of probability per unit of time.

In our computations, we use intensity function

:raw-latex:`\begin{equation} {\mathcal J}(y) = \left\{ \begin{matrix} {\sf r}_1 \left( \exp \left[ {\frac {{\sf r}_2} 2} (y -{\underline y})^2\right] - 1 \right) & y \ge {\underline y} \cr 0 & 0 \le y < {\underline y} \end{matrix} \right. \end{equation}`

Where the values for \(r_1\) and \(r_2\) are as follows:

Parameter	Value
\(r_1\)	1.5
\(r_2\)	2.5

from src.plot import plot_intensity
plot_intensity("Figure 3: Plot of the Intensity Function")